Indestructible colourings and rainbow Ramsey theorems

We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that $2^{ω₁}$ is arbitrarily large, and there is a function g establishing $2^{ω₁} ↛ [(ω₁,ω₂)]_{ω₁}$; but there is no uncountable g-rainbow subset of $2^{ω₁}$. We also show that if GCH holds then for each k ∈ ω there is a k-bounded colouring f: [ω₁]² → ω₁ and there are two c.c.c. posets 𝓟 and 𝒬 such that $V^{𝓟}$ ⊨ f c.c.c.-indestructibly establishes $ω₁ ↛ *[(ω₁;ω₁)]_{k-bdd}$, but $V^{𝒬}$ ⊨ ω₁ is the union of countably many f-rainbow sets.